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Results tagged with homological-algebra
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user 318
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
6
votes
Accepted
a small question about group homology
No, it is not true. For example, let $\mathbf{Z}/2$ act on $\mathbf{Z}$ by inversion, and $G$ be the semidirect product. Then $\mathbf{Z}$ is a torsion-free finite index normal subgroup of $G$, but on …
- 18.2k
1
vote
Equivalence of finiteness of $spliG$ and periodicity isomorphisms being induced by cup product
Let $P_\bullet \to \mathbb{Z}$ be a projective resolution as a $\mathbb{Z}[G]$-module, $\Delta : P_\bullet \to P_\bullet \otimes P_\bullet$ an approximation of the diagonal, and $\phi : P_\bullet \to …
- 18.2k
6
votes
1-st cohomology of multiplicative group in a vector space
By coincidence I needed to know something about this recently, and one thing I know is that that Ext group vanishes for $0 < \vert m - n \vert < p - 1$.
There is a proof in Lemma 6.1 of my paper "Co …
- 18.2k
6
votes
Homology of a limit of semidirect products
No, because it is not even true for constant families: let $A$ be an acyclic group, so $H_i(A)=0$ for $i>0$, and $B$ be a group which $A$ acts on interestingly, e.g. $B= F(A)$ is the free group on the …
- 18.2k
7
votes
Accepted
Is there a kind of Poincare duality for Borel equivariant cohomology?
This kind of thing shows up quite naturally in parameterised stable homotopy theory. Let me translate an idea I know from there into the language in this question.
Cap product gives a map
$$C^{p}(M …
- 18.2k